Definition 2

Let A be a commutative Banach algebra.

Let I be an ideal of A$,x\in A$, and $\chi \in \mathsf{X}'(A)$. One says that $x$ belongs to I in a neighbourhood of $\chi$ if there exists an element $y\in I$ such that $\mathscr{G}'(y)$ and $\mathscr{G}'(x)$ coincide in a neighbourhood of $\chi$.

One says that A satisfies Ditkin's condition if, for every $\chi \in \mathsf{X}'(A)$ and every $x\in A$ such that $\mathscr{G}'(x)$ vanishes at $\chi$, there exists a sequence $(x_n)$ in A such that $x=$ lim$_{n\rightarrow \infty}x_nx$ and such that each $\mathscr{G}'(x_n)$ vanishes in a neighbourhood $V_n$ of $\chi$.